•
-v co
IO
en
If-v
) (
<.D
•
•I
Rigid Woll u=O
Txy=O or v =O
I,,,
•
_ 4x21·75' = 87'
0·6 y/H
0·4
0·2
0·8
0·6 y/H
0·4
0·2 0
0
BONDED F.E. SMOOTH F.E. SMOOTH ANALYTICAL L/H • 1·67
I
\ \
I
\ I
I STATIC I STATIC
I I
I I
I I MODES I MODES
I I I I I I I
I MODE
I
I MOOE
I
I MOOE
0·4 0·8 1·2 0 0·4 0·8 0 0·4 0·8
Pm/YH Pm/YH Pn,m/YH
Figure 5. 5a Castaic example. Static-one-g modal pressure distributions.
BONDED F.E. SMOOTH F.E. SMOOTH ANALYTICAL L/H • 1·67
RMS SUM RMS SUM
ALGEBRAIC SUM ALGEBRAIC SUM -f---L. I
0-4 u~jyH
0·8
Figure 5. Sb
0 0·4 0 0-4
u~j yH
Castaic example. Estimates of maximum earthquake-induced pressures.
Maximum earthquake-induced pressure distributions were computed using the Response Spectrum method, and the 10% damped
*
velocity spectrum given in Appendix VIII. The effects of the higher modes were included by using a ''rigid" mode having a pressure distribution the difference between the appropriate static solution and the sum of the two modal contributions for the bonded wall and the three contributions for both the smooth wall solutions • This
"rigid" mode distribut10n was assumed to contribute at an accele ra- tion of O. 33 g (that is the peak ground acceleration). Plots of the pressure distributions obtained by taking the root-mean-square sums and the algebraic sums of the modal contributions (including the
"rigid" mode) are given in Fig. 5 .Sb. The large difference between the rms and the algebraic sums of the modal pressures for the smooth wall cases is due to the relatively similar magnitude of two of the modal contributions. In the bonded contact case most of the pressure distribution comes from a single mode and thus the differ- ence between the pressures summed by the two methods is relatively small. Near
t
values of 2. 0 it was previously found that therelative magnitude of the modal pressure contributions of the analyti- cal solution were quite sensitive to the value of the
t
parameter.This type of effect is probably the reason for the difference in character of the bonded and smooth contact modal solutions of this problem. For cases in which a significant difference occurs between
*
Because of the closeness of this facility to the San Andreas fault a response spectrum of higher intensity was used for design.the rms and algebraic sum it would appear reasonable to use for design a pressure distribution intermediate between the two results or to at least consider the consequence of an unfavorable combina- tion of the modes.
In general the agreement between the three different solutions is relatively good. Better agreement between the bonded and the smooth contact solutions would be expected for values of equivalent
t
greater than 2. 5 and for equivalentt
values between O. 5 and 1. 5.In these ranges the relative participation of the modal contributions becomes less sensitive to changes in the geometry and boundary conditions .
The random vibration theory method, outlined in Section 3. 8, was used to compute statistical estimates of the earthquake-induced maximum wall force and moment. To obtain approximate equivalence, with the velocity response spectra given in Appendix VIII, the power spectral density shown in Fig. 3. 27 was increased by a factor of 8. O.
(Further study is required to establish a more exact equivalence.) Mean-square forces and moments were computed using this increased power spectral density function and the complex-amplitude response functions given in Fig. 3.19 for the
~
= 2.0, v = 0.4 case. (At
value of 2. 0 was used for this illustration be cause the complex- amplitude responses had been previously computed for this case.) From the properties of the normal distribution curve, a value 2. 5 times greater than the mean will not be exceeded at a probability of 0.988. The P = 0.988 values of wall force and moment are given in Table 5. 2 below. For comparison the earthquake-induced force andmoment were evaluated using the Response Spectrum method. The method was applied by using the three largest modal contributions from the analytical solution for
~ =
2.0 and v=
0.4 (tabulated in Appendix VI), and a "rigid'. mode that was chosen to give a modalsummation equal to the static solutions. The earthquake modal responses were computed from the 10% damped velocity spectrum given in Appendix VIII. Absolute and rms sums of the modal
responses are given in Table 5, 2. The static elastic theory (analytical) force and moment, the Mononobe-Okabe force and moment and vertical gravity values are also given in Table 5. 2.
The vertical gravity force and moment were computed from the analytical expression for the normal stress on a smooth rigid wall given by
= _v_ (1 - Y)
1-v H (5. 1)
where the origin of the coordinate system is assumed at the base of the wall.
The Mononobe-Okabe force and moment are significantly less than the values computed by the other methods. Because the soil is expected to remain essentially elastic for this problem, the
Mononobe-Okabe method is not really applicable; however, the values from this method are given to show the error that might result if the method is used. The force and moment obtained from
the static theory of elasticity solution are probably conservative and this approach appears likely to give a good first approximation for
many rigid wall structures,
TABLE 5. 2
Rigid
Castaic Forces and Moments wall, Smooth Contact, H L
=
2. 0,Method of Computation Random Vibration
Response Spectrum, i;,
=
10%rms sum
Response Spectrum, i;,
=
10%absolute sum
Static Solution for
11, =
0, 33 g (elastic, analytical)Mononobe-Okabe for
11,
= 0, 33 g {triangular pressure distribution) Vertical Gravity(elastic, analytical)
Force/'YH 2 0,140 0,130
0,245
0.235
0.124
0.333
v
= o.
4Moment/')IH 3 0.076 0,071
0.130
0.128
0.041
0.111
6. FORCED WALL
It is convenient to evaluate the earthquake-induced pressures on deformable wall structures by obtaining the solution in two parts;
a rigid-wall solution for soil body forcing, and a solution for dis- placement forcing on the wall boundary. If the wall-soil system is assumed to be linear, the principle of superposition can be applied to combine these two solutions to give the total earthquake-induced pressures. To obtain an essentially exact solution it is necessary to perform the superposition in the frequency domain using the harmonically forced steady-state solutions for the two cases. In this chapter solutions are presented for both static and harmonic displacement forcing on the wall boundary. In Chapter 7 these solu- tions are superimposed with the rigid-wall solutions to give total earthquake forces and moments on the deformable wall.
In Section 1. 2 wall types and their basic deformational behavior under horizontal earthquake loads were discussed. It is clearly not possible to consider all types of wall deformation in a general investigation and in this study only a rotational deformation of the wall about its base is considered in detail. This type of wall displacement will probably be a relatively good approximation for many cases. The methods used for the rotational deformation can be extended to analyze other forms of wall displacement.
6.1. STATIC ANALYTICAL SOLUTION
An analytical solution is presented in this section for the pressures on a smooth wall resulting from a static rotational dis- placement. The soil is· once again as sum e d to be a homogeneous linearly elastic medium. The problem and the boundary conditions are shown in Fig. 6. 1. From the equilibrium equations and the stress-strain relations (expressions (2.1) and (2. 2))
(6. 1)
for O<x<L 0 < y< H
The solution of equatiora (6.1) can be expressed in the following form:
u(x,y) =
v(x,y) =
in which
00
0
)
¥
cos px+ u
(y) sin rx,'--J n
n=O
00
-
~:sin
px+ l v
n (y) cos rxn=O
r = T mr ,n=0,1,2, ...
,,.
P = 2L
u0 = displacement in the x-direction at the top of the wall
(6. 2)
y,v
H
0
Homogenous elastic soil (Plane strain)
~-Rotating